Difference cochain and chain

A difference cochain is an obstruction to the extension of a homotopy between mappings. Let $ X $ be some cellular space, let $ Y $ be a simply-connected topological space and suppose, moreover, that one is given two mappings $ f, g: X \rightarrow Y $ and a homotopy

$$ F \ \mathop{\rm on} ( X \times 0) \cup ( X ^ {n - 1 } \times I ) \cup ( X \times 1) $$

(where $ I = [ 0, 1] $ and $ X ^ {n} $ is the $ n $- dimensional skeleton of $ X $) between these mappings on the $ ( n - 1) $- dimensional skeleton. For every oriented $ n $- dimensional cell $ e ^ {n} $ of $ X $, the restriction of $ F $ to $ \partial ( \overline{e}\; \times I) $ gives a mapping $ S ^ {n} \rightarrow Y $( $ S ^ {n} $ is the $ n $- dimensional sphere) and hence an element of the group $ \pi _ {n} ( Y) $. Thus there arises the cochain $ d ^ {n} ( f, g) \in C ^ {n} ( X; \pi _ {n} ( Y)) $( the notation $ d _ {F} ^ {n} ( f, g) $ would be more precise), which is called the difference cochain; the cochain $ d ^ {n} ( f, g) $ is an obstruction to the extension of $ F $ to

$$ ( X \times 0 ) \cup ( X ^ {n} \times I ) \cup ( X \times 1) = \ ( X \times I) ^ {n - 1 } \cup ( X \times \{ 0, 1 \} ) . $$

The following statements hold: 1) $ d ^ {n} ( f, g) = 0 $ if and only if the homotopy between $ f $ and $ g $ can be extended to $ X ^ {n} $; 2) the cochain

$$ d ^ {n} ( f, g) \in \ C ^ {n} ( X \times I, X \times \{ 0, 1 \} ; \pi _ {n} ( Y)) $$

is a cocycle; 3) the cohomology class

$$ [ d ^ {n} ( f, g)] \in \ H ^ {n} ( X \times I, X \times \{ 0, 1 \} ; \pi _ {n} ( Y) ) $$

vanishes if and only if there is a homotopy between $ f $ and $ g $ on $ X ^ {n} $ that coincides with $ F $ on $ X ^ {n - 2 } $. Without loss of generality one can assume that $ f $ and $ g $ coincide on $ X ^ {n - 1 } $ and that $ F ( x, t) = f ( x) = g ( x) $ for $ x \in X ^ {n - 2 } $. Then the following statements hold:

1) $ d ^ {n} ( f, g) = - d ^ {n} ( g, f ) $, in particular $ d ^ {n} ( f, f ) = 0 $;

2) $ d ^ {n} ( f, g) + d ^ {n} ( g, h) = d ^ {n} ( f, h) $;

3) for any mapping $ f: X \rightarrow Y $ and for any cochain $ d \in C ^ {n} ( X; \pi _ {n} ( Y)) $ there is a mapping $ g $ for which $ f \mid _ {X ^ {n - 1 } } = g \mid _ {X ^ {n - 1 } } $ and $ d ^ {n} ( f, g) = d $.

Now suppose one is given two mappings $ f, g: X ^ {n} \rightarrow Y $, $ f \mid _ {X ^ {n - 1 } } = g \mid _ {X ^ {n - 1 } } $ and let $ c _ {f} ^ {n + 1 } $ and $ c _ {g} ^ {n + 1 } $ be the obstructions to the extensions of the corresponding mappings. The role of the difference cochain in the theory of obstructions is explained by the following proposition:

$$ c _ {f} ^ {n + 1 } - c _ {g} ^ {n + 1 } = \ \delta d ^ {n} ( f, g). $$

Thus, if $ g $ can be extended to $ X ^ {n + 1 } $, then $ [ c _ {f} ^ {n + 1 } ] = 0 $ and if $ [ c _ {f} ^ {n + 1 } ] = 0 $, then $ f \mid _ {X ^ {n - 1 } } $ can be extended to $ X ^ {n + 1 } $.

References

[a1]	G.W. Whitehead, "Elements of homotopy theory" , Springer (1978) pp. 228